No, "bing" by itself does not mean sick. The sound "bing" maps to at least 59 possible characters across the 4 tones[0] of Mandarin. Only one particular character that maps to "bing4" means sick and it's one out of at least 21 possibilities[1], though I'll grant that it's the one that people are most likely to think of first.
Moreover, as it's normally pronounced by westerners, Bing would most likely map to "bing1" anyway.
Sounds more like bing4 to my ear, but see for yourself:
https://upload.wikimedia.org/wikipedia/commons/8/86/Zh-bing1... (https://en.wiktionary.org/wiki/bing1#Mandarin)
https://upload.wikimedia.org/wikipedia/commons/a/a5/Zh-b%C3%... (https://en.wiktionary.org/wiki/b%C3%ACng#Mandarin)